Least variation in tool orientation control for 5-axis CNC machining

We address the problem of minimizing the variation of the tool axis orientation of a 5-axis CNC machine equipped with a ball-end cutter that is designed to maintain a constant cutting speed, which is ensured by a fixed angle ψ between the tool axis a(t) and the workpiece surface normal n(t) at each point of contact as the cutter moves along a specified toolpath. The configuration can be mathematically translated into a pair of curves a(t) and n(t) on the unit sphere with constant parametric distance ψ, and the least variation of the tool axis orientation is achieved when the overall length of the curve a(t) is minimized under certain auxiliary boundary conditions. Using the calculus of variation we show that the solution a(t) consists of arcs of great circles and offset curves of n(t) on the unit sphere. A numerical approximation scheme based on Euler's method of finite differences is also provided. © 2022 Elsevier B.V.